Solving Rotation Problems: Finding Theta and Number of Revolutions

[cde42003]

I have a couple of questions.

1. Assume that the function x(t) represents the length of tape that has unwound as a function of time. Find theta(t), the angle through which the drum will have rotated, as a function of time.

Express your answer (in radians) in terms of x(t) and any other given quantities.

There is a picture for this that I attached below. This question doesn't seem very difficult but I cannot think of a way to solve it. I know that the amount of tape unwound in one revolution is 2*pi*r but do not know how to relate this to x(t)

2. A magnetic computer disk 8.0 cm in diameter is initially at rest. A small dot is painted on the edge of the disk. The disk accelerates at 600 rad/s^2 for .5s, then coasts at a steady angular velocity for another .5s.

A.What is the speed of the dot at t = 1.0 s? This equals 12m/s
b. Through how many revolutions has it turned? This is the problem. I would think that you should find circumference of the circle and use the speed from part a to find the number of revolutions but this has not worked.

Any help would be greatly appreciated. Thanks

 

[OlderDan]

I have a couple of questions.

1. Assume that the function x(t) represents the length of tape that has unwound as a function of time. Find theta(t), the angle through which the drum will have rotated, as a function of time.

Express your answer (in radians) in terms of x(t) and any other given quantities.

There is a picture for this that I attached below. This question doesn't seem very difficult but I cannot think of a way to solve it. I know that the amount of tape unwound in one revolution is 2*pi*r but do not know how to relate this to x(t)

2. A magnetic computer disk 8.0 cm in diameter is initially at rest. A small dot is painted on the edge of the disk. The disk accelerates at 600 rad/s^2 for .5s, then coasts at a steady angular velocity for another .5s.

A.What is the speed of the dot at t = 1.0 s? This equals 12m/s
b. Through how many revolutions has it turned? This is the problem. I would think that you should find circumference of the circle and use the speed from part a to find the number of revolutions but this has not worked.

Any help would be greatly appreciated. Thanks

1) 2*pi*r is x(t) when t is the time for one revolution. More generally x(t) at any time is the length of the arc subtending the angle theta. Do you know the definition of theta in radian measure?

2) During constant acceleration, the angular velocity is constantly changing. If that acceleration is constant, the average angular velocity is the average of the final and initial angular velocities. You need to use the average to figure out how far the disk rotated while accelerating.

 

[cde42003]

1) 2*pi*r is x(t) when t is the time for one revolution. More generally x(t) at any time is the length of the arc subtending the angle theta. Do you know the definition of theta in radian measure?

2) During constant acceleration, the angular velocity is constantly changing. If that acceleration is constant, the average angular velocity is the average of the final and initial angular velocities. You need to use the average to figure out how far the disk rotated while accelerating.

Thanks for the help. I figured both questions out now.

 

[OlderDan]

The definition of an angle in radian measure is the arc length subtending the angle divided by the radius. In your diagram, the arc length between the horizontal reference line and the line that terminates angle theta is the length of tape that has unrolled, x(t). By definition, the angle theta is that arc length divided by the radius.

 

[Hippo]

The circumference of a circle is

$$= \theta r$$,

where $$\theta$$ is measured in radians. There are, of course, 2$$\pi$$ radians 'bout a full circle.


[Edit: Sorry for the redundancy. I didn't notice OlderDan's post.]